Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C25×#442 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C4×D8 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,17,12,19,30,18)(7,20,24,36,26,14,25,35)(8,27,22,13,28,33,23,34)(9,29,21,15,10,31,32,16);(2,4)(5,7,11,24,12,26,30,25)(6,13,17,33,19,34,18,27)(8,21,22,10,28,32,23,9)(14,16,35,29,20,15,36,31);(5,8)(6,14)(7,21)(9,25)(10,24)(11,22)(12,28)(13,16)(15,34)(17,35)(18,36)(19,20)(23,30)(26,32)(27,31)(29,33)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12939, #494294, #494295, … |
| maximal normal subgroup of | C2×#12939, #494294, #494295, … |
| maximal quotient of | #135881, #135882, #135883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5927);